Order of Dissipation Near Rarefraction Centers

We consider the approximation of weak solutions of hyperbolic systems of conservation laws, 1.1ut + f(u)x = 0, - backslashinfty < x < backslashinfty,t >0;u( backslashtimes,0)given,by projection methods of finite-difference, finite-element, spectral, etc. type (as opposed to methods such as those of Godunov [2] or Glimm [1]). These methods all contain a dissipation term or mechanism, as needed for the generation of entropy in the presence of shocks. (Throughout this discussion, we specialize to systems for which there exists a convex entropy function U, with corresponding entropy flux F [3].) In the presence of shocks, the magnitude of the required dissipation is determined by the requirement that entropy be generated at the correct rate, given a discrete shock profile uniformly bounded (i.e. without excessive overshooting) and confined to a width of 0(h), where h is the mesh size. For example, a regularized form of (1.1) such as 1.2ut + f(u)x = h(A(u,hux)ux)xif often used on the basis of a discretization procedure, for a suitably chosen viscosity matrix A(textperiodcentered,textperiodcentered).